First off, welcome aboard. That is a very detailed first post that was fun to read.
Mrbb does have a potential point. Having said that, squirrel hunting can be all over the map on season, bag limits, etc. Since you posted tree rats, I'm assuming you're not talking about ground squirrels but the kind that tend to climb and live in trees a lot. Just doing a quick search, it looks like Virginia and New Hampshire have several month long squirrel hunting seasons with pretty liberal bag limits (5 or 6 per day). Texas apparently has 157 counties where squirrel hunting is open year round, another 51 counties in Eastern Texas that have both spring and fall squirrel seasons and the panhandle that doesn't allow squirrel hunting. All of this, of course, assumes that you live in the US. We had a similar topic a while back where the member actually lived in another country with completely different game laws. So if it's legal where you live, enjoy the time with your dog. If it's not legal, obviously discontinue hunting and bone up on your state's hunting laws. Again, welcome aboard.